The Poincar\'e Recurrence Problem of Inviscid Incompressible Fluids

Nadirashvili presented a beautiful example showing that the Poincar\'e recurrence does not occur near a particular solution to the 2D Euler equation of inviscid incompressible fluids. Unfortunately, Nadirashvili's setup of the phase space is not appropriate, and details of the proof are missing. This note fixes that

On the so-called rogue waves in the nonlinear Schr\"odinger equation

The mechanism of a rogue water wave is still unknown. One popular conjecture is that the Peregrine wave solution of the nonlinear Schr\"odinger equation (NLS) provides a mechanism. A Peregrine wave solution can be obtained by taking the infinite spatial period limit to the homoclinic solutions. In this article, from the perspective of the phase space structure of these homoclinic orbits in the infinite dimensional phase space where the NLS defines a dynamical system, we exam the observability of these homoclinic orbits (and their approximations). Our conclusion is that these approximate homoclinic orbits are the most observable solutions,and they should correspond to the most common deep ocean waves rather than the rare rogue waves. We also discuss other possibilities for the mechanism of a rogue wave: rough dependence on initial data or finite time blow up

Symbolic dynamics and chaos in plane Couette flow

According to a recent theory \cite{Li14}, when the Reynolds number is large, fully developed turbulence is caused by short term unpredictability (rough dependence upon initial data); when the Reynolds number is moderate, often transient turbulence is caused by chaos (long term unpredictability). This article aims at studying chaos in plane Couette flow at moderate Reynolds number. Based upon the work of L. van Veen and G. Kawahara \cite{VK11} on a transversal homoclinic orbit asymptotic to a limit cycle in plane Couette flow, we explore symbolic dynamics and chaos near the homoclinic orbit. Mathematical analysis shows that there is a collection of orbits in the neighborhood of the homoclinic orbit, which is in one-to-one correspondence with the collection of binary sequences. The Bernoulli shift on the binary sequences corresponds to a chaotic dynamics of a properly defined return map

The distinction of turbulence from chaos -- rough dependence on initial data

I propose a new theory on the nature of turbulence: when the Reynolds number is large, violent fully developed turbulence is due to "rough dependence on initial data" rather than chaos which is caused by "sensitive dependence on initial data"; when the Reynolds number is moderate, (often transient) turbulence is due to chaos. The key in the validation of the theory is estimating the temporal growth of the initial perturbations with the Reynolds number as a parameter. Analytically, this amounts to estimating the temporal growth of the norm of the derivative of the solution map of the Navier-Stokes equations, for which here I obtain an upper bound $e^{C \sqrt{t Re} + C_1 t}$. This bound clearly indicates that when the Reynolds number is large, the temporal growth rate can potentially be large in short time, i.e. rough dependence on initial data.Comment: arXiv admin note: text overlap with arXiv:1305.286

Chaos in Partial Differential Equations, Navier-Stokes Equations and Turbulence

I will briefly survey the most important results obtained so far on chaos in partial differential equations. I will also survey progresses and make some comments on Navier-Stokes equations and turbulence

Invariant Manifolds and Their Zero-Viscosity Limits for Navier-Stokes Equations

First we prove a general spectral theorem for the linear Navier-Stokes (NS) operator in both 2D and 3D. The spectral theorem says that the spectrum consists of only eigenvalues which lie in a parabolic region, and the eigenfunctions (and higher order eigenfunctions) form a complete basis in $H^\ell$ ($\ell = 0,1,2, ...$). Then we prove the existence of invariant manifolds. We are also interested in a more challenging problem, i.e. studying the zero-viscosity limits (\nu \ra 0^+) of the invariant manifolds. Under an assumption, we can show that the sizes of the unstable manifold and the center-stable manifold of a steady state are $O(\sqrt{\nu})$, while the sizes of the stable manifold, the center manifold, and the center-unstable manifold are $O(\nu)$, as \nu \ra 0^+. Finally, we study three examples. The first example is defined on a rectangular periodic domain, and has only one unstable eigenvalue which is real. A complete estimate on this eigenvalue is obtained. Existence of an 1D unstable manifold and a codim 1 stable manifold is proved without any assumption. For the other two examples, partial estimates on the eigenvalues are obtained.Comment: 28p

On the True Nature of Turbulence

In this article, I would like to express some of my views on the nature of turbulence. These views are mainly drawn from the author's recent results on chaos in partial differential equations \cite{Li04}. Fluid dynamicists believe that Navier-Stokes equations accurately describe turbulence. A mathematical proof on the global regularity of the solutions to the Navier-Stokes equations is a very challenging problem. Such a proof or disproof does not solve the problem of turbulence. It may help understanding turbulence. Turbulence is more of a dynamical system problem. Studies on chaos in partial differential equations indicate that turbulence can have Bernoulli shift dynamics which results in the wandering of a turbulent solution in a fat domain in the phase space. Thus, turbulence can not be averaged. The hope is that turbulence can be controlled.Comment: 7 page

Recurrence in 2D Inviscid Channel Flow

I will prove a recurrence theorem which says that any $H^s$ ($s>2$) solution to the 2D inviscid channel flow returns repeatedly to an arbitrarily small $H^0$ neighborhood. Periodic boundary condition is imposed along the stream-wise direction. The result is an extension of an early result of the author [Li, 09] on 2D Euler equation under periodic boundary conditions along both directions

Rough dependence upon initial data exemplified by explicit solutions and the effect of viscosity

In this article, we present some explicit solutions showing rough dependence upon initial data. We also studies viscous effects. An extension of a theorem of Cauchy to the viscous case is also presented

Arnold Diffusion of the Discrete Nonlinear Schr\"odinger Equation

In this article, we prove the existence of Arnold diffusion for an interesting specific system -- discrete nonlinear Schr\"odinger equation. The proof is for the 5-dimensional case with or without resonance. In higher dimensions, the problem is open. Progresses are made by establishing a complete set of Melnikov-Arnold integrals in higher and infinite dimensions. The openness lies at the concrete computation of these Melnikov-Arnold integrals. New machineries introduced here into the topic of Arnold diffusion are the Darboux transformation and isospectral theory of integrable systems.Comment: 24p